INVARIANT SUBSPACES IN THE POLYDISK

Transcription

1 PACIFIC JOURNAL OF MATHEMATICS Vol. 121, No. 1, 1986 INVARIANT SUBSPACES IN THE POLYDISK O. P. AGRAWAL, D. N. CLARK AND R. G. DOUGLAS This note is a study of unitary equivalence of invariant subspaces of H 2 of the polydisk. By definition, this means joint unitary equivalence of the shift operators restricted to the invariant subspaces. In one variable, all invariant subspaces are unitarily equivalent and all can be represented as inner functions times H 2. In several variables, our results suggest that unitary equivalence and multiplication by inner functions are again related. For example, all invariant subspaces of a given invariant subspace Jt which are unitarily equivalent to Jί are φjΐ, for φ inner; and all invariant subspaces unitarily equivalent to an invariant subspace Ji of finite codimension are yjί. In particular, two invariant subspaces of finite codimension are unitarily equivalent if and only if they are equal. 0. The classic paper of Beurling [4] led to a spate of research in operator theory, H p theory, and other areas, which continues to the present. Although Beurling's answer to the problem of spectral synthesis was negative, his explicit characterization of the invariant subspaces of the unilateral shift, in terms of the inner-outer factorization of analytic functions has had a major impact. Since the Hardy space for the unit disk is the primary nontrivial example for so many different areas, it is not surprizing that this characterization has proved so important. Almost everyone who has thought about this topic must have considered the corresponding problem for H 2 of the polydisk. Although the existence of inner functions in this context is obvious (in contrast with the case of the ball), one quickly sees that a Beurling-like characterization is not possible. Most results on this problem have gone unpublished [10], except for the book of Rudin [11], and many results are counterexamples. One exception is a characterization, by Ahern and the second author, of invariant subspaces having finite codimension, as the closure of the ideals in C[z v...,z N ] of finite codimension; [2]. In this note, we eschew the goal of characterizing all invariant subspaces in the polydisk and instead investigate equivalence of invariant subspaces in N variables, under joint unitary equivalence of the restrictions of the N coordinate shifts. Put another way, we shall consider the problem of characterizing the (inequivalent) submodules of the Hardy space over the polydisk algebra. The existence of nonunitarily equivalent submodules is directly related to the

2 2 O. P. AGRAWAL, D. N. CLARK AND R. G. DOUGLAS failure of inner functions to characterize invariant subspaces; [7]. We believe that this may be a worthwhile approach to this problem. A discussion of this point of view has been published by M. Cowen and the third author in [8]. The first result on nonunitarily equivalent invariant subspaces is due to Berger, Coburn and Lebow [3], who considered the restrictions of multiplication by the coordinate functions to the invariant subspaces obtained as the closure of certain ideals in C[z v z 2 ] having the origin as zero set. By applying their results on commuting isometries, they showed that different ideals yield inequivalent invariant subspaces. In [7], Cowen and the third author reproved these results using their complex geometric approach to operator theory. In his Dissertation [1], the first author extended these methods to cover the invariant subspaces defined by most ideals in C[z v z 2 ] with zero set consisting of an arbitrary point in the bidisk. This proof is indicated in 5 below. The results of [1] led us to realize that all ideals in C[z v z 2,...,z N ] of finite codimension define nonunitarily equivalent invariant subspaces of H 2. We present two distinct proofs of this result in 3 and 4, along with a number of questions. The different proofs generalize in different directions. 1. Let Z denote the integers, Z + the nonnegative integers and N a fixed positive integer greater than 1. For a subset A of Z N, we identify I 2 (A) with the Hubert space of Fourier series with ll/ll 2 = Σ aga \a a \ 2 < oo, where e iat = e' * e ia»'». Such a function / can be viewed as defined on the N torus T^; this correspondence defines the usual isometrical isomorphism between 1 2 (Z N ) and L 2 (Ύ N ). The subspace / 2 (Z+) corresponds to the Hardy space H 2 (Ύ N ). Each function φ in L (Ύ N ) determines on L 2 (Ύ N ) a multiplication operator M φ9 defined by M φ f=φf. When φ e H (Ί N ), the operator M φ leaves H 2 (Ύ N ) invariant, and by an invariant subspace of H 2 (Ύ N ), we mean a closed subspace of H 2 (Ύ N ) 9 invariant under M φ for φ e H 00 ; or, equivalently, invariant under the shifts S x = Me ifl,... 9 S N = Me ιtn. Two invariant subspaces Jί γ and Jί' 2 are said to be unitarily equivalent if there is a unitary operator U: Jί x -> Jΐ 2 such that U(ψf) = φ(uf) 9 for φ e H 0C (Ύ N ) and/ e JK V This is equivalent to joint unitary equivalence of the restrictions S λ \^,...,S N \^, fory = l,2.

3 INVARIANT SUBSPACES IN THE POLYDISK 3 The above notion of equivalence can also be viewed as equivalence of submodules of H 2 (Ύ N ) over the polydisk algebra ^4(D^) (D the open unit disk). Here we are using submodule to mean closed subspace of H 2 (Ύ N ), invariant under the action of ^(D^). The natural notion of unitary equivalence of modules then becomes unitary equivalence of invariant subspaces, as defined above. We begin with a lemma which exhibits the form a unitary equivalence of invariant subspaces must have. LEMMA 1. If Jt x andjί 2 are invariant subspaces of H 2 (Ύ N ) and if U: Jί λ -> Jt' 2 is a unitary operator making the restricted shifts jointly unitarily equivalent USnU^S^U, n = l,...,n, then there exists a unimodular function φ Ξ L CO (Ύ N ) 9 such that Uf=φf, f^jί λ. Proof. Since Jί λ and Jt 2 are invariant subspaces, the restrictions of S S N to Jt λ and Jt 2 are commuting N-tuples of isometries, the minimal unitary extensions of which are both S l S N on L 2 (Ύ N ) (the proof is the same as that in the one variable case; cf. [5, Theorem 12]). Now a straightforward application of a corollary in [9] shows that U can be extended to a unitary operator W on L 2 (Ύ N ) that commutes with the multiplication operators {S n }% =1. Setting φ = W\, we obtain, for f ~ Σα α e ιαt = L 2 (Ύ N \ that Wf= WLa a S?»l Spi = Σα α e ιαt Wl = φf. Thus W is multiplication by φ and φ is unimodular since W is unitary. Now given two invariant subspaces Jt λ and Jt' 2 which are unitarily equivalent, there are several possibilities for the function φ of Lemma 1: (a) φ is constant; (b) φ is inner; (c) φ is the quotient of relatively prime inner functions; (d) φ is the quotient of inner functions; or (e) φ is not the quotient of inner functions. When (a) holds, Jt Ύ = Jt 2 \ w e obtain some results in this direction. Note that since, on T^, there exist nontrivial inner functions, i.e. nonconstant, unimodular functions in H 2 (Ύ N ), it follows that there exist distinct unitarily equivalent invariant subspaces. λίjtis an invariant subspace and φ is an inner function, then yjl c ^with equality occurring if and only if ψ is constant; in fact φjf c.jί'xί and only if ψ is inner. This is proved in Proposition 3 below. A consequence is that the unitary operator U of Lemma 1 is unique up to a constant factor of modulus 1.

4 4 O. P. AGRAWAL, D. N. CLARK AND R. G. DOUGLAS The statement that the inner functions ψ λ and φ 2 are relatively prime means that whenever Φi/i = Φ2/2 holds for/ 1? f 2 in H 2 (Ύ N ), it follows that If (c) holds lhεnjί ι and^# 2 can be represented as (1) Jί x for the subspace An apparently weaker condition is (d), where the inner functions φ λ and φ 2 are not assumed to be relatively prime. In that case, the space Jί defined above is an invariant subspace of L 2 (Ύ N ) 9 but not of H 2 (Ύ N ). Expressing Jί x and Jt\ in terms of Jί as in (1), with φ x and φ 2 inner is equivalent to (c) if Jί can be taken to be an invariant subspace of H 2 (Ύ N ) or (d) if it is an invariant subspace of L 2 (Ύ N ). An interesting question is to give conditions under which an invariant subspace of L 2 (Ύ N ) is unitarily equivalent to an invariant subspace of H 2 (Ύ N ). The problem of whether an arbitrary unitarily equivalent pair of invariant subspaces has a representation (1) is equivalent to the following problem concerning the modulus of an H 2 (Ύ N ) function. If/! and/ 2 are functions in H 2 (Ύ N ) satisfying \f λ \ = / 2 a.e. on T", do there exist inner functions φ x and <p 2 such that/^ = f 2 φ{l To see the equivalence in one direction, observe that if φjί 1 = Jί 2 and if f γ is a nonzero function in Jί x c H 2 (Ύ N ), then φf λ =/ 2 is a nonzero function in Jί 2 c H 2 (Ύ N ). Since φ = f λ /f 2 we see that if a representation as a quotient of inner functions is possible for f 1 /f 2, then it is possible for φ. Conversely, if such a representation is not possible for fι/f 2, then the cyclic invariant subspaces Jί λ &ndjί 2 generated by/ x and f 2 are unitarily equivalent but have no representation of the form (1), since the function φ given by Lemma 1 in this case is just f 2 /f v Added in proof. Case (d) (without (c)) and case (e) can, in fact, occur. Examples have recently been given by W. Rudin [J. Funct. Analysis, 61 (1985), ]. 2. We now set down two different sets of hypotheses under which we have made progress on concluding that unitarily equivalent invariant subspaces must be equal. The first is geometrical in nature, while the second is function theoretic. Both cover the case in which the invariant subspaces have finite codimension in H 2 (Ύ N ).

5 INVARIANT SUBSPACES IN THE POLYDISK 5 we have An invariant subspace^of / 2 (Z+) has full range if, for n = 1,2,...,N 9 (2) \f Sf j Jί= 1 2 (ZI~ 1 X Z X Z?- π ). 7 = 1 Note that for any subspace ~# of / 2 (Z+) and any n we have inclusion of the left side of (2) in the right side. PROPOSITION 1. If Jί is a full range invariant subspace then any invariant subspace unitarily equivalent to Jί is of the form JV= φ^#, for some inner function φ. Proof. In the notation of Lemma 1, we have JV= φjf, for some unimodular function φ. Thus we have K v s: 7 = 1 and since Jί has full range, it follows that = V S?M ψ Jl= \/ Sϊ 7-1 7=1 A/ φ (/ 2 (ZΓ 1 X Z x Z?-")) c / 2 (ZΓ X x Z x 1 N -"). Taking intersection over n = 1,2,...,N gives which implies that φ = M φ l e H 2 (Ύ N ), and hence is inner. COROLLARY 1. An invariant subspace Ji is unitarily equivalent to H 2 (Ύ N ) if and only ifjί = φh 2 (Ύ N )for some inner function ψ. COROLLARY 2. If' Jί x andjί 2 are full range invariant subspaces, then they are unitarily equivalent if and only if they are equal. COROLLARY 3. If Jί γ andjί 2 are invariant subspaces of finite codimension in i/ 2 (T /v ), then they are unitarily equivalent if and only if they are equal. Proof. Proposition 1 applies since a subspace of finite codimension has full range. REMARK 1. This result should be compared with the situation in case N = 1, in which all nonzero invariant subspaces of H 2 (Ύ) are unitarily equivalent.

6 6 O. P. AGRAWAL, D. N. CLARK AND R. G. DOUGLAS REMARK 2. The result of Corollary 3 remains true if H 2 (Ύ N ) is replaced by an invariant subspace^of H 2 (Ύ N ) (see Proposition 4 below). REMARK 3. Let {φ k } be a sequence of inner functions on T satisfying ψ k H 2 c φ k+ι H 2 ΐoτk = 0,1,2,... and pj k H 2 = H 2. If we let Jί be the subspace 00 then Jί is a full range invariant subspace which has infinite codimension unless some φ^ is a constant. If {ψ^} is another sequence of inner functions on T having the same properties, then the span of the doubly infinite sequence of subspaces is a more complex example of the same phenomenon. If ~/Tis an invariant subspace of H 2 (Ύ 2 ) and φ(e itι, e'* 2 ) is a nonconstant inner function, then φ^γis an invariant subspace not having full range. It would be interesting to know if every invariant subspace not having full range can be written in this way. 3. Our function theoretic approach depends upon knowing that certain kinds of functions lie in the subspaces. PROPOSITION 2. Let Jί be an invariant subspace such that, for each n, 1 < n < N, Jί contains a function independent of e ιtn. Then every invariant subspace unitarily equivalent to Jί is of the form yjί, for some inner function φ. Proof. With φ as in Lemma 1, write 00 If p is a nonzero function in Jί which is independent of t υ we have that φp isin H 2 (Ύ N ). Since φp has the expansion φp~ Σ

7 INVARIANT SUBSPACES IN THE POLYDISK 7 this implies pf. = 0 for j < 0 and hence fj = O for j < 0. Treating the expansions of φ relative to the other variables e itn, n = 2,3,...,iV, shows that φ is in H 2 (Ύ N ) and hence is inner. COROLLARY 4. If' J( λ and Jί 2 are invariant subspaces, both of which contain functions independent of e ιί % for n = 1,2,...,7V, then Jί λ and Jt' 2 are unitarily equivalent if and only if they are equal. REMARK 4. Of course, Corollaries 1 and 3 can also be derived from Proposition 2; the latter because if Jί has finite codimension in H 2 (Ύ N ), then Jί always contains the functions (e«-z ι ) k,(e">-z 2 ) k,...,(e''»-z N ) k for some (z l9 z 2,... 9 z N ) e Ό N and a sufficiently large integer k\ [2, Theorem 3]. 4. Carrying our polydisk function theory ideas a little further allows us to obtain the promised result on inclusion of invariant subspaces. PROPOSITION 3. Two invariant subspaces Jί λ andjί 2 satisfying Jί\ c Jί λ are unitarily equivalent if and only if Jί' 2 = ψjί λ, for some inner function φ. Proof. Unitary equivalence of M x and ψjί λ is obvious. Conversely, if Jί λ 2inάJΐ 2 are unitarily equivalent, φ is the unimodular function given by Lemma 1, and / is a nonzero function in Jί λ, then φ k f is in H 2 (Ύ N ) for k = 1,2, Using an idea of Schneider [12], we shall prove this implies φ e i/ (T N ). We set g = φf and extend φ to those 2 in D^ where f(z) Φ 0 by defining φ(z) = g(z)/f(z). To prove the extended function φ is bounded and holomorphic in D^, we first claim that φ(z) < 1 at any point z inό N where/(z) Φ 0. To prove this claim, set h k = φ k f. Since we have h k (e") = φ(e") k f(e«) = g ( * " 7 * \ f(eηh k (e ιt ) = φ{e ιt ) k - l f{e«)g{e><) = Since/, h k, h k _ λ and g lie in H 2 (D N ), both/tz^ and h k _ x g are in H ι (D N ) and we have (by the Cauchy formula, for example) that for z in D N, f{z)h k {z) = h k _ ι {z)g{z)

8 8 O. P. AGRAWAL, D. N. CLARK AND R. G. DOUGLAS or, by induction, that for z in D",/(z) Φ 0. Now since \h k (e")\ = /(<?")l a.e. we have for every z = (z 1,...,z A,)inD iv, where c = Πftl - \Zj\ 2 )~ 1/2 and hence if z is in D* and/(z) φ 0, Since the right hand side is independent of k, this proves φ(z) < 1. Now the fact that φ(z) = g(z)/f(z) is holomorphic in D^ follows from Hartogs' theorem. Thus φ is holomorphic in D^, bounded in modulus by 1 and unimodular on T^, which proves that φ is inner. The next proposition gives the generalization of Corollary 3 referred to in Remark 2. PROPOSITION 4. If' Jί x andjt 2 are unitarily equivalent invariant subspaces contained in an invariant subspace Jί and if' Jt x has finite codimension injί, thenjί 1 = φ*jf v for some inner function φ. Proof, As might be expected, the proof combines the ideas of the proofs of Propositions 2 and 3. Let φ be the unimodular function given by Lemma 1. We claim that φvlies in H 2 (Ύ N ), for every/in^and for k = 0,1,2,... Of course the claim is correct if k = 0. Now proceed by induction, assuming φ k ~ ι f belongs to H 2 (Ύ N ), for every/in Jί. \ίjί x has codimension m in Jt, the functions /, e ιtχ f, e ι2tl f,...,e ιmtι f lie in Jί and so some linear combination Σcje ιjh f' = p γ (e ιtχ )f belongs to Jί γ, It follows that φ/? 1 (e //l )/lies injί 2 and hence in^#, and therefore by the induction hypothesis. Repeating the argument with t λ replaced by r 2, ί 3,... 9 t N9 we obtain N polynomials p λ (e itι ), p 2 (e it2 ),...,p N (e itn ) such that for n = l,2,...,iv, belongs to H 2 (Ύ N ). The proof of Proposition 2 shows that this implies φ k f belongs to H 2 (Ύ N ). Now the proof of Proposition 3 shows that φ is inner.

9 INVARIANT SUBSPACES IN THE POLYDISK 9 5. We conclude this note by offering another proof that invariant subspaces defined by a certain class of finite codimensional ideals in C[z v z 2 ] have the property that they are unitarily equivalent if and only if they are equal. The methods used from complex geometry offer the possibility of a much more detailed analysis of these invariant subspaces than the results obtained above. This is the reason we include them here. Recall that in [6] a certain class of commuting pairs of operators was studied. A pair 2Γ= (τ l9 T 2 ) of bounded linear operators on the Hubert space ^is said to belong to the class ^(Ω), for Ω an open connected subset of C 2, if the following conditions are satisfied: (1) ran D^_ x is closed for λ in Ω, where is defined by D^x = T λ x Θ T 2 x and ran D*r is the range of (3) dimker D^_ λ = 1, for λ e Ω. To each ^ i n ^(Ω) is associated the Hermitian holomorphic line bundle Undefined by E^= {(λ, x) eβx/:χg ker2v_ λ }. In [6] it is proved that the curvature of this bundle is a complete unitary invariant for the pair &~. The relevance of this to the study of invariant subspaces Jί of H 2 (Ύ N ) is that the operator pairs (Sf, S ) 9 where now S λ and S 2 are the restrictions tojίoi multiplication by e ih and e 1 * 2, respectively belongs to ^(Ω) for some Ω c D 2 for many (and perhaps all) invariant subspaces Jί. Thus if the curvature of the associated line bundle can be calculated, then one can decide when two invariant subspaces are unitarily equivalent. Let 0 < p x < p 2 < - < p r and 0 < q r < q r x < < q λ be integers and let (λ 1? λ 2 ) be a point in D 2. Let J p q λ be the ideal in C[z v z 2 ] consisting of those polynomials p(z) for which the Taylor series expansion P(z)-Σa Jk (z 1 -λ 1 ) J (2 2 -λ 2 ) k, about the point (λ l9 λ 2 ), has a jk = 0 if j < p i and k < q t for some 1 < / < r. The closure Jί p qλ oίj pqλ in iί 2 (T 2 ) is an invariant subspace of finite codimension. Let &~ p%q% \ denote the operator pair y= (Sf, S 2 *),

10 10 O. P. AGRAWAL, D. N. CLARK AND R. G. DOUGLAS defined above, for this invariant subspace. Another proof of the fact that the~# x are nonunitarily equivalent, for distinct/?, q, λ, can be based on the following PROPOSITION 5. The pair 3~ p^χ is in ^(D 2 \ {λ}) and the curvature Ctif xfor the associated line bundle is given by 2 = Σ dω- (l- Wl 2 )(l- co 2 2 ) F 3ω 3ω n Λ dω k, vhere F p.aλ<*><*) = r+1 λ 1 ω 1 ω 2 - λ 2 λ 2 co 2 ω 2 - λ 2 andq r+ι = -1. 1=1 1 λ 2 ω 2ω 2 Proof. One can reduce the proof of SΓ p q λ G 33 λ to the case (λ 1? λ 2 ) = (0,0) using a pair of Mδbius transformations on D and then apply the result for this case proved in [7, p. 20]. The curvature calculation is done using the orthonormal basis for H 2 (Ί), a part of which forms a basis ίoτjί pq χ. COROLLARY 5. The invariant subspaces Jί' andjί rs β are unitarily equivalent if and only if p = r, q = s and(a v a 2 ) = (β v β 2 ). Proof. By definition, Jί r and pqa Jίf r s β are unitarily equivalent if and only if ^p^a and 3F r^ are. Since these operator pairs both belong to β 1 (D 2 \{a, β}), their unitary equivalence reduces to a comparison of curvatures. By inspection, both curvatures are real analytic functions which extend to an open neighborhood of the closed bidisk. Fixing one of the variables on the unit circle, it is possible, by comparing Taylor coefficients as in [1], to show that equality of the curvatures implies that p = r, q = s and (α 1? α 2 ) = (β l9 β 2 ). We will not go into details.

11 INVARIANT SUBSPACES IN THE POLYDISK 11 As mentioned earlier, the curvature contains encoded all the data for the invariant subspace, up to unitary equivalence, and should prove useful in a more detailed study. We hope to return to this at some later date. REFERENCES [I] O. P. Agrawal, Invariant subspaces of shift operator for the quarter plane, PhD Thesis, SUNY Stony Brook, [2] P. R. Ahern and D. N. Clark, Invariant subspaces and analytic continuation in several variables, J. Math. Mech., 19 (1970), [3] C. A. Berger, L. A. Coburn and A. Lebow, Representation and index theory for C*-algebras generated by commuting isometries, J. Functional Analysis, 27 (1978), [4] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math., 81 (1949), [5] A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew, Math., 213 (1963), [6] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math., 141 (1978), [7], On operators possessing an open set of eigenvalues, Proc. Fejer-Riesz conference, Budapest, [8], On moduli for invariant subspaces, in Invariant subspaces and other topics (Timisoara/Herculane, 1981) pp , Operator Theory: Adv. Appl. 6, Birkhaiiser, Basel-Boston, Mass., [9] R. G. Douglas, On extending commutative semigroups of isometries, Bull. London Math. Soα, 1 (1969), [10] H. Helson, Lectures on Invariant Subspaces, Academic Press, New York, [II] W. Rudin, Function Theory in Polydisks, Benjamin, New York, [12] R. B. Schneider, Isometries ofh p (U"), Canad. J. Math., 25 (1973), Received May 2, The second and third authors were supported by NSF grants. UNIVERSITY OF KANSAS LAWRENCE, KS UNIVERSITY OF GEORGIA ATHENS, GA AND SUNY STONY BROOK, NY 11794

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